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Let $so_3(K)$ be the Lie algebra of $3\times 3$ skew-symmetric matrices overa field $K$ of characteristic 0. The ideal $I(M_3(K),so_3(K))$ of the weakpolynomial identities of the pair $(M_3(K),so_3(K))$ consists of the elements$f(x_1,\ldots,x_n)$ of the free associative algebra $K\langle X\rangle$ withthe property that $f(a_1,\ldots,a_n)=0$ in the algebra $M_3(K)$ of all $3\times3$ matrices for all $a_1,\ldots,a_n\in so_3(K)$. The generators of$I(M_3(K),so_3(K))$ were found by Razmyslov in the 1980's. In this paper thecocharacter sequence of $I(M_3(K),so_3(K))$ is computed. In other words, the${\mathrm{GL}}_p(K)$-module structure of the algebra generated by $p$ genericskew-symmetric matrices is determined. Moreover, the same is done for theclosely related algebra of $\mathrm{SO}_3(K)$-equivariant polynomial maps fromthe space of $p$-tuples of $3\times 3$ skew-symmetric matrices into $M_3(K)$(endowed with the conjugation action). In the special case $p=3$ the latteralgebra is a module over a $6$-variable polynomial subring in the algebra of$\mathrm{SO}_3(K)$-invariants of triples of $3\times 3$ skew-symmetricmatrices, and a free resolution of this module is found. The proofs involvemethods and results of classical invariant theory, representation theory of thegeneral linear group and explicit computations with matrices.

Cocharacters for the weak polynomial identities of the Lie algebra of $3\times 3$ skew-symmetric matrices